Cascading of up conversion and down conversion

ABSTRACT

In digital video, 3:4 down conversion and 4:3 down conversion are transparently cascaded. The up conversion filter is: S up (n)=ΣS input (k).g(3n−4k) and the down conversion filter is: S down (n)=ΣS up (k).h(4n−3k). The pair of up and down conversion filters are designed so that Σh(4n−3k).g(3k−4m) is equal to unity if n=m and is otherwise equal to zero.

FIELD OF THE INVENTION

This invention relates to digital filtering of sampled signals and in animportant example to filtering of digital video

BACKGROUND

The sampling rate of a digital video has to be changed for convertingbetween different formats. For example, in order to save bandwidth avideo signal may be stored and transmitted in a low resolution formatbut displayed in an up converted high resolution format. Typically, thesampling rate of the low resolution is related to the sampling rate ofthe high resolution by a ratio 1:M or by a ratio N:M where N and M areintegers. Depending on the application, several cascades of up and downconversion filtering can occur. For example in a broadcast chain this isdue to repeated editing, preview and storage operations. In order toachieve a perfect reconstruction of those parts of the pictures thathave not been changed by editing operations up conversion from the lowresolution to the high resolution and subsequent down conversion shouldresult in a transparent cascade. Transparent cascading of N:M up and M:Ndown conversion filtering imposes additional problems on the filterdesign when compared with a cascade of up and down conversion filtersfor ratios 1:M and M:1.

SUMMARY

It is an object of one aspect of the present invention to provide animproved digital filtering process capable of transparent up conversionand cascaded down conversion.

Accordingly, the present invention consists in one aspect in a digitalfiltering process for achieving a transparent cascade on N:M upconversion and subsequent M:N down conversion, where the up and downconversion ratios N:M and M:N respectively are rational numbers and theintegers N and M satisfy the condition 1<N<M, wherein the up conversionfilter operates on a sampled signal S_(input) and is chosen to take theform s_(up)(n)=Σs_(input)(k).g(Nn−Mk) and wherein a corresponding downconversion filter operates on the up converted signal s_(up) and ischosen to take the form s_(down)(n)=Σs_(up)(k).h(Mn−Nk); the pair (g,h)of up and down conversion filters being chosen so thatΣh(Mn−Nk).g(Nk−Mm) is equal to unity if n=m and, is otherwise equal tozero.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will now be described by way of example with reference tothe accompanying drawings, in which

FIG. 1 is a diagram showing the low resolution and high resolutionsampling grid for 1:2 up conversion and subsequent 2:1 down conversion;

FIG. 2 is a diagram showing the low resolution and high resolutionsampling grid for 3:4 up conversion and subsequent 4:3 down conversionand showing, additionally, the sampling grid for the intermediateresolution;

FIG. 3 is a block diagram illustrating the three stages of 3:4 upconversion, that is to say 1:4 up sampling, interpolation filtering, and3:1 down sampling;

FIG. 4 is a block diagram illustrating the three stages of 4:3 downconversion, i.e. that is to say 1:3 up sampling, interpolationfiltering, 4:1 down sampling; and

FIG. 5 is a graph showing the frequency response of 3:4 up conversionfilter, bilinear and proposed filter.

DETAILED DESCRIPTION

For a ratio 1:M and isochronous sampling, the sampling grid of the lowresolution signal forms a subset of the sampling grid of the highresolution signal as shown in FIG. 1 for the example ratio 1:2. Thus, ifthe samples of the low resolution signal are copied onto thecorresponding positions of the high resolution grid on 1:M up conversionfiltering, one subset that contains every M-th sample of the upconverted signal coincides with the set of samples of the low resolutionsignal. An up conversion filter that has this property is called a I:MNyquist filter. Therefore, if a 1:M Nyquist filter is followed by a M:1down sampler one obtains mathematically the identity operator whichmakes the cascade transparent.

However, this approach does not work for a ratio N:M such as 3:4. Thereason is that the sampling grid of the low resolution signal is not anentire subset of the sampling grid of the high resolution signal. Forexample in the case of 3:4 up conversion only every third grid positionof the low resolution signal coincides with a corresponding gridposition of the high resolution signal as shown in FIG. 2. Thus, onlyone third of the samples can be copied from the low to the highresolution. Consequently, only one third of the samples of the lowresolution signal can be recovered simply by sub-sampling the upconverted signal. Additional filtering is needed on down conversion torecover the remaining two-thirds.

An example of a filtering process according to the present inventionwill now be described. This will for illustration concentrate on theexample ratios 3:4 on up conversion and 4:3 on down conversion.

First, some principles of up and down conversion filtering are brieflyreviewed. The sampling grids of the low and the high resolution areshown in FIG. 2 for 3:4 up conversion and subsequent 4:3 downconversion. Additionally, FIG. 2 shows the intermediate sampling grid onwhich all filter operations are performed. It is assumed throughout thatall filter operations are linear and shift invariant. Thus, each filtercan be described by its impulse response. Let g denote the impulseresponse of the 3:4 up conversion filter. The 3:4 up conversion processcan be considered in three stages, see FIG. 3.

In the first stage the low resolution signal s is up sampled by thefactor 1:4 resulting in the signal t which is defined on theintermediate sampling grid as follows, $\begin{matrix}{{t\quad (n)} = \left\{ \begin{matrix}{{s\quad (k)\quad {if}\quad n} = {4k}} \\{0\quad {otherwise}}\end{matrix} \right.} & (1)\end{matrix}$

In the second stage the signal t is interpolated into the signal u,$\begin{matrix}{{u\quad (n)} = {\sum\limits_{k}^{\quad}\quad {t\quad {(k) \cdot g}\quad \left( {n - k} \right)}}} & (2)\end{matrix}$

In the third stage the interpolated signal u is down sampled by thefactor 3:1, resulting in the up converted signal

s _(up)(n)=u(3n)  (3)

Equations (1) to (3) define the functional relationship between the lowresolution signal s and the up converted signal s_(up), $\begin{matrix}{{s_{up}\quad (n)} = {\sum\limits_{k}^{\quad}\quad {s\quad {(k) \cdot g}\quad \left( {{3n} - {4k}} \right)}}} & (4)\end{matrix}$

As every third position of the low resolution sampling grid coincideswith a position of the high resolution sampling grid, the samples ofthese positions can be copied on 3:4 up conversion,

s _(up)(4n)=s(3n)  (5)

One can derive equation (5) from equation (4) if the filter g fulfilsthe Nyquist condition for 1:4 up conversion, $\begin{matrix}{{g\quad \left( {4n} \right)} = \left\{ \begin{matrix}1 & {{{if}\quad n} = 0} \\0 & {otherwise}\end{matrix} \right.} & (6)\end{matrix}$

In this description a 3:4 up conversion filter g that fulfils equation(6) is called a 3:4 Nyquist filter. Correspondingly, equation (5) iscalled the 3:4 Nyquist condition.

Similar to 3:4 up conversion, the 4:3 down conversion process can beconsidered in three stages, see FIG. 4. The relationship between the upconverted signal s_(up), and the down converted signal s_(down) becomes$\begin{matrix}{{s_{down}\quad (n)} = {\sum\limits_{k}^{\quad}\quad {s_{up}\quad {(k) \cdot h}\quad \left( {{4n} - {3k}} \right)}}} & (7)\end{matrix}$

where h denotes the impulse response of the 4:3 down conversion filter.

Similar to equation (5), the Nyquist condition for 4:3 down conversioncan be specified,

s _(down)(3n)=s _(up)(4n)  (8)

One can derive equation (8) from equation (7) if the filter h fulfilsthe Nyquist condition for 1:3 up conversion, $\begin{matrix}{{h\quad \left( {3n} \right)} = \left\{ \begin{matrix}1 & {{{if}\quad n} = 0} \\0 & {otherwise}\end{matrix} \right.} & (9)\end{matrix}$

In this description a 4:3 down conversion filter h that fulfils equation(9) is called a 4:3 Nyquist filter.

One concludes from equations (5) and (8) that the cascade of a 3:4Nyquist filter which is followed by a 4:3 Nyquist filter gives a perfectreconstruction for every third sample of the low resolution signal, i.e.

s _(down)(3n)=s(3n)  (10)

However, in order to recover the remaining two-thirds, resulting in

s _(down)(n)=s(n)  (11)

the pair (g, h) of up and down conversion filters has to fulfil thecondition: $\begin{matrix}{{\sum\limits_{k}^{\quad}\quad {h\quad {\left( {{4n} - {3k}} \right) \cdot g}\quad \left( {{3k} - {4m}} \right)}} = \left\{ \begin{matrix}1 & {{{if}\quad n} = m} \\0 & {otherwise}\end{matrix} \right.} & (12)\end{matrix}$

Equation (12) is the condition for transparent cascading that can bederived straightforwardly from equations (4) and (7). It shows that thefilter coefficients of g and h cannot be chosen independently. For givencoefficients of g equation (12) becomes a linear equation system for thecoefficients of h and vice versa.

From equation (12) one can derive the corollary, that if equation (12)is fulfilled by the pair of up and down conversion filters (g,h) and,additionally, the impulse response of g is symmetric, i.e. g(n)=g(−n)then:

i) equation (12) is also fulfilled by the pair of filters (g, h_) withh_(n)=h(−n)

ii) equation (12) is also fulfilled by the pair of filters (g, h˜) withh˜=(1−λ).h+λ.h_ and ˜∞≦λ≦∞, in particular the impulse response ofh˜=(h+h_)/2 is symmetric.

The up converted signal s_(up) should give a good picture quality ondisplaying. Therefore, firstly the coefficients of the up conversionfilter g can be chosen to fulfil this requirement and then acorresponding down conversion filter h can be designed to comply withequation (12), resulting in a transparent cascade.

The length of the impulse response is an obvious limitation in practicalapplications. Bilinear interpolation results in a short impulseresponse. The coefficients of the symmetric impulse response g_(bil) areshown in Table 1. However, bilinear interpolation filtering does notresult in a sophisticated low pass characteristic. Therefore, Table 1shows also the symmetric impulse response of the filter g_(def) that isproposed for 3:4 up conversion. The filter coefficients are derived froma windowed sin(x)/x waveform to give a better low pass characteristicthan the bilinear interpolation filter. The frequency responses ofg_(bil) and g_(def) are compared in FIG. 5. The improved low passcharacteristic comes at a price as 15 taps are needed for g_(def) butonly 7 taps for g_(bil). The coefficients of g_(def) are rounded to sixdecimal digits as shown in Table 1. It is not difficult to verify thatboth up conversion filters of Table 1 are 3:4 Nyquist filters.

For the given 3:4 Nyquist filters g_(bil) and g_(def) corresponding 4:3Nyquist filters h_(undo-bil) and h_(undo-def) respectively, are designedin order to comply with equation (12). The filter coefficients are alsolisted in Table 1. Similar to g_(def) the coefficients of h_(undo-def)are rounded to six decimal digits. In contrast to the up conversionfilters, the corresponding down conversion filters of Table 1 do nothave a symmetric impulse response. Again, the impulse response length ofthe proposed filter is larger, i.e. 29 taps are needed for h_(undo-def)but only 9 taps for h_(undo-bil).

It follows from the corollary to equation (12) that the non-symmetricimpulse responses of the down conversion filters of Table 1 can beconverted into symmetric impulse responses by mirroring. However, thiswould increase the number of filter taps.

It will be understood that this invention has been described by way ofexample only and that a wide variety of modifications are possiblewithout departing from the scope of the invention.

It has already been explained that the ratios 3:4 and 4:3 have beenchosen by way of illustration only. The invention is more broadlyapplicable to cascaded N:M up conversion and subsequent M:N downconversion, where the up and down conversion ratios N:M and M:Nrespectively are rational numbers and the integers N and M satisfy thecondition 1<N<M. In the field of video, the invention is applicable notonly to horizontal processing, but also to vertical processing andtemporal processing. If processing is required in two or more of thesedimensions, it will generally be possible to identify pairs of onedimensional filters in accordance with the present invention and then tocascade those filters to achieve the desired two or three dimensionalprocessing.

TABLE 1 Nyquist filter for 3:4 up conversion, bilinear (bil) andproposed filter (def) and Nyquist filter for subsequent 4:3 downconversion, undo bilinear (undo-bil), undo proposed (undo-def) 3:4 upconversion n g_(bil)(n) n h_(undo-bil)(n) n g_(def)(n) n h_(undo-def)(n)0 1 −8  1/3 0 1 −8 0.003449 −1, 1 3/4 −7 0 −1, 1 0.854892 −7 0 −2, 2 1/2−6 0 −2, 2 0.530332 −6 0 −3, 3 1/4 −5 −4/3 −3, 3 0.204804 −5 0 otherwise0 −4 −1/3 −4, 4 0 −4 0.053745 −3 0 −5, 5 −0.054860 −3 0 −2 2 −6, 6−0.030332 −2 0.113710 −1  4/3 −7, 7 −0.004832 −1 0 0 1 otherwise 0 0 1otherwise 0 1 1.099232 2 1.771901 3 0 4 −0.222855 5 −1.099232 6 0 70.005838 8 0.280050 9 0 10 0.000604 11 −0.005838 12 0 13 0 14 −0.00060415 0 16 0.000018 17 0 18 0 19 0 20 −0.000018 otherwise 0

What is claimed is:
 1. A filtering process for achieving a transparentcascade on up conversion in the sampling rate ratio N:M where N and Mare integers and subsequent M:N down conversion, where the up and downconversion ratios N:M and M:N respectively are rational numbers and theintegers N and M satisfy the condition 1<N<M, wherein the up conversionfilter operates on a sampled signal S_(input) and is chosen to take theform S_(up)(n)=ΣS_(input)(k).g(Nn−Mk) where k is the running integerover which the sum is taken and wherein a corresponding down conversionfilter operates on the up converted signal Sup and is chosen to take theform S_(dowm)(n)=ΣS_(up)(k).h(Mn−Nk); the pair (g, h) of up and downconversion filters being chosen so that Σh(Mn−Nk).g(Nk−Mm) is equal tounity if n=m and is otherwise equal to zero.
 2. A process according toclaim 1, wherein the up conversion filter is a N:M Nyquist filter with asymmetric impulse response and the down conversion filter is a M:NNyquist filter with a non-symmetric impulse response.
 3. A processaccording to claim 1, wherein the up conversion filter is a N:M Nyquistfilter with a symmetric impulse response and the down conversion filteris a M:N Nyquist filter with a symmetric impulse response.
 4. A pair ofrespectively N:M up conversion and subsequent M:N down conversionfilters, where the up and down conversion ratios N:M and M:Nrespectively are rational numbers and the integers N and M satisfy thecondition 1<N<M, wherein the up conversion filter is adapted to operateon a sampled signal s_(input) and is chosen to take the form ofs_(up)(n)=Σs_(input)(k).g(Nn−Mk) and wherein the down conversion filteris adapted to operate on the up converted signal s_(up) and is chosen totake the form of s_(down)(n)=Σs_(up)(k).h(Mn−Nk); the pair (g,h) of upand down conversion filters being chosen so that Σh(Mn−Nk).g(Nk−Mm) isequal to unity if n=m and is otherwise equal to zero.
 5. A M:N downconversion filter adapted to operate on the output s_(up)(n) of a N:M upconversion digital filter of the form S_(up)(n)=Σs_(input)(k). g(Nk−Mn),where the up and down conversion ratios N:M and M:N respectively arerational numbers and the integers N and M satisfy the condition 1<N<M,wherein the down conversion filter is chosen to take the forms_(down)(n)=Σs_(up)(k).h(Mn−Nk); where Σh(Mn−Nk).g(Nk−Mm) is equal tounity if n=m and is otherwise equal to zero.